Signal processor, signal processing method, signal processing program, recording medium with the signal processing program recorded therein and measuring instrument

ABSTRACT

A digital signal value is divided to a plurality of zones along a route. A median is computed based on a difference between a digital signal value and a sum of squares of each component of a filter output value for the digital signal values for each zone. A weighting factor for digital signals in each zone is computed and updated based the median for the zone. A filter output value for the digital signal value is obtained by executing filtration using the computed weighting factor.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a signal processor, a signal processingmethod, a signal processing program, a recording medium with the signalprocess program recorded therein, and a measuring instrument. Morespecifically this invention relates to a signal processor, a signalprocessing method, a signal processing program each for executing theprocessing for filtering measurement data obtained two-dimensionally orthree-dimensionally, a recording medium with the signal processingprogram recorded therein, and a measuring instrument.

2. Description of Related Art

There have been known various types of surface characteristics measuringinstruments for measuring a contour, roughness, waviness and otherproperties of a surface of a workpiece for measurement such as acoordinate measuring machine for measuring a three-dimensional form of aworkpiece for measurement, a contour measuring instrument or an imagemeasuring instrument for measuring a two-dimensional contour, aroundness measuring instrument for measuring roundness, and a surfaceroughness measuring instrument for measuring waviness, roughness orother properties of a surface of a workpiece for measurement. Thesemeasuring instruments are used for collection of measurement dataconcerning a surface of a workpiece for measurement by moving a contacttype sensor or a non-contact type sensor and the workpiece formeasurement relatively to each other.

The measurement data collected as described above generally includesexternal disturbance factors such as noises.

The external disturbance factors mainly include electric/magneticinduction noises including high frequency factors, and when it isnecessary to measure a contour of a surface of a workpiece formeasurement, such factors as surface roughness and waviness may be theexternal disturbance factors.

In order to remove the external disturbance factor according to thenecessity, there has been employed the method in which measurement datais converted from analog signals to digital signals and the digitalsignals are subjected to filtering with a filtering program for acomputer. By means of this filtering processing, for instance, the highfrequency factors are removed.

There has been known the Gaussian regression filter as a filter capableof executing filtering as described above (Refer to, for instance,reference 1: ISO/TR 16610-10:2000(E) Geometrical Product Specification(GPS)-Data extraction techniques by sampling and filtration-Part 10:Robust Gaussian regression filter, 1999). This filter executes theGaussian distribution function type of weighting to measurement datay_(i) (I=1, 2, 3, . . . ) obtained when a surface of a workpiece formeasurement is measured with a prespecified sampling pitch Δx in thex-axial direction.

Assuming the Gaussian distribution function as s_(ik), the filter outputg_(k) for the measurement data y_(i) is expressed as follows:

$\begin{matrix}{{g_{k} = {{\sum\limits_{i = 0}^{n - 1}\;{{y_{i} \cdot s_{ik}^{\prime}}\mspace{25mu} k}} = 0}},1,2,{{\cdots\mspace{11mu} n} - 1}} & (1)\end{matrix}$

Herein the Gaussian distribution function s_(ik) is standardized and isexpressed by the following expression:

$\begin{matrix}{{s_{ik} = {{\frac{1}{\lambda_{c}\sqrt{\ln(2)}} \cdot \exp}\left\{ {- \frac{{\pi^{2} \cdot \left( {i - k} \right)^{2} \cdot \Delta}\; x^{2}}{{\ln(2)} \cdot \lambda_{c}^{2}}} \right\}}}{s_{ik}^{\prime} = \frac{s_{ik}}{\sum\limits_{i = 0}^{n - 1}s_{ik}}}} & (2)\end{matrix}$wherein Δx indicates a sampling pitch along the x-axis and λ_(c)indicates a cut-off wavelength.

Further there has been known the robust Gaussian regression filterhaving the robustness provided by adjusting a weighting factor for eachmeasurement data according to a degree of residual error d_(k) betweenthe measurement data y_(i) and the data g_(k) having been subjected tofiltration. (Refer to, for instance 1, reference 2: S. Brinkmann et al.,Accessing roughness in three-dimensions using Gaussian regressionfiltering, International Journal of Machine Tools & Manufacture 41(2001) 2153-2161, reference 3: S. Brinksmann et al., Development of arobust Gaussian regression filter for three-dimensional surfaceanalysis, Xth International Colloquium on Surface, 2000, pp 122-132).

With the Gaussian regression filter and the robust Gaussian regressionfilter as described above, all data can be subjected to filtrationwithout the necessity of deleting some of measurement data or addingquasi data. Especially filtration can be carried out suppressinggeneration of distortion at both ends of a measurement area.

Further, with the robust Gaussian regression filter, a result offiltration can be obtained without being affected by abnormal data.

In the robust Gaussian regression filter updating a weighting factor,there has been known the method in which a weighting factor is updatedbased on a residual error between each data value and an output valueafter the initial processing for filtration, and in updating a weightingfactor, each data is weighted based on a median of the residual error.

However, when one median is obtained for all of measurement data andthis median is applied to all of the data, local fluctuations of thedata can not be grasped.

For instance, in a case of data with a very small noise level as shownin FIG. 11, a median of a residual error between initial data and filteroutput is very small. Therefore when a weighting factor is updatedaccording to the median, a number of data regarded as abnormal data andhaving the apparent weight of zero increases. Especially, distortionsare generated at both ends of the measurement area due to the initialprocessing, and therefore the distortions become very larger by therobust processing associated with updating of a weighting factor.

Further, in a case where there is a step-formed abrupt change in thedata as shown in FIG. 12, data becomes dull by a value equivalent to thecut-off wavelength through the initial processing. Therefore, when aweight is updated according to the median, a weight of data becomes zeroat a step-like changing point in the data, and as a result the filteroutput waveform does not reflect an accurate form of a measured surfacebecause of the robust processing updating a weight.

SUMMARY OF THE INVENTION

An object of the present invention is to provide a signal processor anda signal processing method making it possible to obtain a filter outputvalue accurately following data by filtration processing using anappropriate weighting factor, a signal processing program for themethod, a recording medium with the signal processing program recordedtherein, and a measuring instrument.

The signal processor according to the present invention executesfiltration processing for digital signal values at a prespecifieddimension measured along a preset route, and includes a weighting factorcalculating section for calculating a weighting factor for each of thedigital signal values and also for re-calculating and updating theweighting factor, and a filtration executing section for obtaining afilter output value for each of the digital signals by executingfiltration using the weighting factor calculated in the weighting factorcalculating section, and the weighting factor calculating sectionincludes a first zone generating section for generating a plurality offirst zones by dividing the digital signal values to a plurality of zonealong the route, and a median calculating section for calculating amedian for a difference based on a sum of squares of a residual error ineach component between the digital signal value and the filter outputvalue for the digital signal value for each of the first zones, and theweighting factor calculating section calculates and updates theweighting factor based on the median in each of the first zones for thedigital signal values in the first zone.

With the configuration as described above, a digital signal value isdivided to a plurality of zones (first zones) by the first zonegenerating section, and a median is calculated for each zone by theintermediate calculating section. The median calculated for each zone isapplied to the zone, and a weighting factor for digital signal valuesfor the zone is calculated and updated by the weighting factorcalculating section. Filtration is executed by the filtration executingsection based on the weighting factor obtained as described above. Withthis configuration, as a weighting factor for digital signal valuesinputted for each zone can be obtained appropriately, filter outputvalues accurately following forms of all digital signal values can beobtained.

For instance, when all of relatively similar signal values are processedin batch and a median is applied to all of the digital signal values,even a signal value slightly off from the basic value is treated asabnormal data, and the data does not contribute to the filter outputvalue at all, but in this invention, weighting for signal values isjudged for each zone, so that filter outputs each accurately reflectingfeatures of each zone can be obtained.

The signal processor according to the present invention executesfiltration processing for digital signal values at a prespecifieddimension measured along a preset route, and includes a weighting factorcalculating section for calculating a weighting factor for each of thedigital signal values and also for re-calculating and updating theweighting factor, and a filtration executing section for obtaining afilter output value for each of the digital signals by executingfiltration using the weighting factor calculated in the weighting factorcalculating section, and the weighting factor calculating sectionincludes a first zone generating section for generating a plurality offirst zones by dividing the digital signal values to a plurality of zonealong the route, a second zone generating section for generating aplurality of second zones corresponding to the first zones for eachdigital value along the route, and a median calculating section forcalculating a median for a difference based on a sum of squares of aresidual error in each component between the digital signal value andthe filter output value for the digital signal value for each of thesecond zones, and the weighting factor calculating section calculatesand updates the weighting factor based on the median for each of thesecond zones corresponding to the first zones respectively.

With the configuration as described above, a digital signal value isdivided to a plurality of zones (first zones) by the first zonegenerating section. Further second zones corresponding to the firstzones respectively are generated by the second zone generating section.The second zones correspond to the first zones respectively, each of thesecond zones are set, for instance, so that a median suitable forjudging weighting for signal values in the first zone can be calculated.Then a median is calculated for the second zone by the mediancalculating section. Weighting for each signal value in thecorresponding first zone is executed based on this median.

By defining a zone (second zone) for judging weighting for each signalvalue independently from a zone (first zone) to which the calculatedmedian is applied, accurate judgment can be made for correctly weightingeach signal value in each first zone. As a result, each signal value ineach zone (first zone) can be weighted more accurately, so that filteroutput values more accurately reflecting features of all inputteddigital values can be obtained.

In the present invention, the second zone generating section preferablygenerates the second zones each having a zone width larger than that ofthe first zone.

With the configuration as described above, second zones each reflectingnot only signal values in the first zone but also those around the firstzone are set. Because of this feature, the obtained filter output valuessufficiently reflect feature of each zone (first zone) based on a mediancalculated for each zone (second zone) and also fully reflect thetendency in all of the signal values.

For instance, when a median based on only signal values within a firstzone is calculated and applied to digital signal values, even signalvalues which should originally be regarded as abnormal values are notdeleted, and in that case the general situation may not accurately bereflected by the filter output values. With the present invention,however, as signal values in the first zone are weighted based on amedian reflecting not only situation in the first zone but also thosearound the first zone, the general tendency can always be graspedaccurately.

In the present invention, the first zone generating section preferablygenerates the first zones so that the digital signal values foradjoining sections are continuously connected to each other and thesecond zone generating section preferably generates the second zonesincluding digital signal values for at least the corresponding firstzone and having overlapping portions between adjoining zones.

With the configuration as described above, each first zone iscontinuously connected to adjoining zones, so that the filter outputvalues corresponding to all of the digital signal values can be obtainedby connecting the filter output vales for the adjoining first zones.Further as each of the second zones include portions overlapping theadjoining zones, so that the median calculated for each second zonereflects also change in signal values for the preceding and followingzones. As a result, it is possible to obtain filter output values not sooff from adjoining values and sufficiently reflecting the generaltendency.

In the present invention, the filtration executing section preferablyexecutes the robust spline filtration processing using the splinefunction.

In the present invention, the filtration executing section preferablyexecutes the robust Gaussian filtration processing using the Gaussianfunction.

With the configuration as described above, a robust spline filter or arobust Gaussian filter capable of providing the effects according to thepresent invention may be employed. With the robust spline filter, it ispossible to suppress generation of distortions at edge portions ofdigital signal values and also to secure the tracking capability forwaviness components having a long cycle. With the robust Gaussianfilter, the processing, for instance, for adding quasi data to edgeportions of digital signal values is not required, and it is possible toobtain filter output values effectively suppressing distortion at twoedge portions of digital signal values.

The signal processing method according to the present invention enablesexecution of filtration processing for digital signal values at a presetdimension measured along a prespecified route, and the method includes aweighting factor calculating step of calculating a weighting factor foreach of the digital signal values and also re-calculating and updatingthe weighting factor; and a filtration executing step of obtainingfilter output values for the digital signal values using the weightingfactor calculated in the weighting factor calculating section, and theweighting factor calculating step includes a first zone generating stepof generating a plurality of first zones by dividing the digital signalvalues to a plurality of zones along the route, and a median calculatingstep of calculating a median for a difference based on a sum of squaresof a residual error in each component between the digital signal valueand the filter output value for the digital signal value for the digitalsignal value for each of the first zones, and the weighting factorcalculating step is executed for calculating and updating the weightingfactor for the digital signal values for each of the first zones basedon the median for the first zone.

The signal processing method according to the present invention enablesexecution of filtration processing for digital signal values at a presetdimension measured along a prespecified route, and the method includes aweighting factor calculating step of calculating a weighting factor foreach of the digital signal values and also re-calculating and updatingthe weighting factor; and a filtration executing step of obtainingfilter output values for the digital signal values using the weightingfactor calculated in the weighting factor calculating section, and theweighting factor calculating step includes a first zone generating stepof generating a plurality of first zones by dividing the digital signalvalues to a plurality of zones along the route, and the weighting factorcalculating section includes a first zone generating step of generatinga plurality of first zones by dividing the digital signal values to aplurality of zones along the route, a second zone generating step ofgenerating a plurality of second zones corresponding to the first zonesrespectively along the prespecified route for the digital signal values,and a median calculating step of calculating a median for a differencebased on a sum of squares of a residual error in each component betweenthe digital signal value and the filter output value of the digitalsignal value for each of the second zones, and further the weightingfactor calculating step is executed for calculating and updating theweighting factor based on the median for each of the second zonescorresponding to the first zones respectively for the digital signalvalues in each of the first zones.

With the configuration as described above, the same effects andadvantages as those provided by the invention described above areprovided.

The signal processing program according to the present invention makes acomputer execute the signal processing method according to the presentinvention.

The recording medium according to the present invention records thereinthe signal processing program according to the present invention.

With the configuration as described above, the same effects andadvantages as those provided by the present invention can be achieved.Further with the configuration in which the program is executed step bystep by a computer incorporating therein a CPU (Central Processing Unit)or a memory (storage device), parameters in each step can easily bechanged.

For instance, a pitch for dividing a digital signal value to a pluralityof first zones or a width of a second zone can easily be changed, andtherefore adjustment can easily be performed so that filter outputvalues accurately reflecting the digital signal values can be obtained.

The program may be installed in a computer by directly inserting arecording medium with the program recorded therein into the computer, ora reader capable of reading information from a recording medium isexternally connected to a computer so that the program is read out andinstalled in this computer from the reader. Further the program may besupplied via the Internet, LAN cable, a telephone line, a radiocommunication line to a computer for installation therein.

The measuring instrument according to the present invention includes adetector having a measuring section for scanning a surface of aworkpiece to be measured in the contact state or in the non-contactstate at its tip section, a moving unit for two-dimensionally orthree-dimensionally moving the measuring section, a position detectingsection for outputting a result of measurement by the measuring sectionas coordinate measurement data, a movement controlling section forinstructing movement of the measuring section, the signal processoraccording to the present invention, and an output section for outputtinga result having been subjected to filtration by the signal processor.

With the configuration as described above, form analysis can be carriedout by fetching data concerning a form of a workpiece to be measurementby scanning a surface of the workpiece for measurement and furtherprocessing the measurement data.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a view showing configuration of a coordinate measuring machineaccording to a first reference technology according to the presentinvention;

FIG. 2 is a view showing a result of the Gaussian filtration totwo-dimensional data in which a sinusoidal wave with the amplitude of 1mm is overlaid on a circle with the radium of 10 mm;

FIG. 3A and FIG. 3B are views each showing the Gaussian filtration totwo-dimensional data in which Gauss noise with the standard deviation of0.1 mm is overlaid on a logarithmic spiral, and FIG. 3A shows a casewhere the cut-off wavelength is 0.5 mm, while FIG. 3B shows a case wherethe cut-off wavelength is 1.0 mm;

FIG. 4A and FIG. 4B are views each showing a result of Gaussianfiltration to two-dimensional data in which the Gauss noise with thestandard deviation of 0.1 mm is overlaid on a folium, and FIG. 4A showsa case where the cut-off wavelength is 0.5 mm, while FIG. 4B shows acase where the cut-off wavelength is 1.0 mm;

FIG. 5A and FIG. 5B are views each showing a result of Gaussianfiltration to two-dimensional data in which Gauss noise with thestandard deviation of 0.1 mm is overlaid on design data for a air foil,and FIG. 5A shows a case where the cut-off wavelength is 0.5 mm, whileFIG. 5B shows a case where the cut-off wavelength is 1.0 mm;

FIG. 6 is a functional block diagram for a computing section based on asecond reference technology according to the present invention;

FIG. 7 is a flow chart showing the steps of robust Gaussian regressionfor measurement data in the second reference technology above;

FIG. 8A and FIG. 8B are views each showing a result of robust Gaussianfiltration for two-dimensional data, and FIG. 8A shows a case where datawith spike noise added to a folium is processed with the cut-offwavelength of 0.1 mm, while FIG. 8B shows a case where data with spikenoise added to design data for an air foil is processed with the cut-offwavelength of 0.5 mm;

FIG. 9A and FIG. 9B are views each showing comparison between filtrationusing the Beaton-Function based on ISO and the robust Gaussianfiltration using the adapted type biweight method according to thepresent invention;

FIG. 10 is a view showing a box-type function as a reference variant ofthe distribution function;

FIG. 11 is a view showing a result of initial processing and robust-likesignal processing for data containing few noises;

FIG. 12 is a view showing a result of initial processing and robust-likesignal processing for data containing step-like changes;

FIG. 13 is a view showing the situation in which the robust-like signalprocessing is being performed by dividing a signal to a plurality ofzones each having a prespecified pitch in a first embodiment of thepresent invention;

FIG. 14 is a view showing the situation in which a range for computingmedian data is different from a range of signal values to be processedwith the median data when the robust-like signal processing is beingperformed by dividing a signal to a plurality of zones each having aprespecified pitch in the first embodiment of the present invention;

FIG. 15 is a view showing a case where the first embodiment of thepresent invention is applied to two-dimensional data;

FIG. 16 is an enlarged view showing a case where the first embodiment ofthe present invention is applied to two-dimensional data;

FIG. 17 is a view showing the situation in which the first embodimentabove is applied to data containing step-like changes;

FIG. 18 is a flow chart showing the signal processing sequence in athird reference technology relating to the signal processing methodaccording to the present invention;

FIG. 19 is a functional block diagram showing a device for signalprocessing in the third reference technology above;

FIG. 20A and FIG. 20B are views each showing comparison between a resultof spline processing for one-dimensional time series data and a resultof robust spline processing for the same data in the third referencetechnology above;

FIG. 21A and FIG. 21B are views each showing comparison between a resultof spline processing and a result of robust spline processing in a fifthreference technology according to the preset invention;

FIG. 22 is a view showing the transfer characteristic according to areference technology for the present invention; and

FIG. 23 is a flow chart showing a reference variant of the presentinvention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENT(S)

Embodiments of the present invention are described below with referenceto the drawings.

The signal processor and signal processing method according to thepresent invention mainly relate to a method of computing a weightingfactor in the robust processing, and as a presumption for description ofthe present invention, at first the Gaussian filter and robust Gaussianfilter are described below as reference technologies.

First Reference Technology

FIG. 1 shows a coordinate measuring machine as a measuring instrumentaccording to a first reference technology for the present invention.

This coordinate measuring machine 1 includes a stylus (detectingsection) 2 having a probe (measuring section) 21 for scanning a surfaceof a workpiece 10 to be measured in the contact or non-contact stateprovided at a tip thereof, a moving unit 3 for moving the probe 21three-dimensionally in the x-axial, y-axial, and z-axial directions; aposition detecting section 4 for detecting a position of the probe 21and outputting data on the position as measurement data, a controlsection 5 for giving instructions for movement of the probe 21 and alsocomputing the measurement data, and a display section 6 as an outputsection for displaying a result of computing.

The control section 5 includes a movement control section 51 foroutputting a control signal for moving the probe 21 along a surface ofthe workpiece 10 to be measured to the moving unit 3, and a computingsection 52 as a signal processor for computing measurement data.

The computing section 52 includes a Gaussian regression filter(filtration executing section) 521 as a signal processing filter.Coordinates of the probe 21 detected by the position detecting section 4are inputted as measurement data into the computing section 52, and thecomputing section 52 subjects the measurement data to the Gaussianregression.

Generally the measurement data is defined as plane curve datadistributed over a plane in a three-dimensional space or as space curvedata three-dimensionally distributed in a three-dimensional space, butherein for convenience of description, of the data obtained throughmeasurement of or defined as an orthogonal coordinate system havingthree axes of axis x, axis y, and axis z, plane curve data on the x-yplane having a constant z-axial coordinate value is defined astwo-dimensional data. The space curve data three-dimensionally presentin this orthogonal coordinate system is defined as three-dimensionaldata.

The two-dimensional data can be obtained, for instance, by clamping thez-axial shaft of a three-dimensional measurement instrument and movingonly the x-axial and y-axial shafts thereof to move the probe 21 over asurface of a workpiece to be measured and executing sampling with aprespecified pitch Δ1. In this case the measurement data has a constantz-axial coordinate value, so that the measurement data can be expressedas (x_(i), y_(i)) for convenience.

Two-dimensional data may also be obtained as measurement data (x_(i),y_(i)) by reading a curve drawn on a plane with a scanner and readingcoordinate values with a prespecified pitch Δ1 on the orthogonal x-yplane.

The computing section 52 executes the Gaussian regression to themeasurement data (x_(i), y_(i)) as two-dimensional data. The filteroutput (g_(x,k), g_(y,k)) by the computing section 52 is expressed usingthe Gaussian distribution function with the following expression;

$\begin{matrix}{{{\begin{matrix}{g_{x,k} = {\sum\limits_{i = 0}^{n - 1}{x_{i} \cdot s_{ik}^{\prime}}}} \\{g_{y,k} = {\sum\limits_{i = 0}^{n - 1}{y_{i} \cdot s_{ik}^{\prime}}}}\end{matrix}k} = 0},1,2,{{\cdots\mspace{11mu} n} - 1}} & (3)\end{matrix}$

When the sampling pitch is the length Δ1 along a measurement route andthe cut-off wavelength is a wavelength λ′_(c) also long the measurementroute, the Gaussian distribution function s′_(ik) is expressed by thefollowing expressions:

$\begin{matrix}{{s_{ik} = {{\frac{1}{\lambda_{c}^{\prime}\sqrt{\ln(2)}} \cdot \exp}\left\{ {- \frac{{\pi^{2} \cdot \left( {i - k} \right)^{2} \cdot \Delta}\; l^{2}}{{\ln(2)} \cdot \lambda_{c}^{\prime 2}}} \right\}}}{s_{ik}^{\prime} = \frac{s_{ik}}{\sum\limits_{i = 0}^{n - 1}s_{ik}}}} & (4)\end{matrix}$

Namely the computing section 52 outputs a sum of products of theGaussian distribution function s′_(ik) and measurement data obtained bymoving along the measurement route by computing the inputtedtwo-dimensional data with the expression (3).

Operations of the coordinate measuring machine 1 having theconfiguration as described above for subjecting the measurement data toGaussian regression are described below.

At first, a surface of a workpiece to be measure is scanned with theprobe 21 keeping the z coordinate at a constant value. In this step,coordinate positions of the probe 21 are sampled by the positiondetecting section 4 with a prespecified pitch along the route formeasurement, and the sampled measurement data (x_(i), y_(i)) is sent tothe computing section 52. The measurement data (x_(i), y_(i)) issubjected to filtration by the Gaussian regression filter 521 in thecomputing section 52. Namely, the filter output (g_(x,k), g_(y,k))having been subjected to the processing with the expression (3) isobtained. The obtained filter output is displayed on the display section6.

FIG. 2, FIG. 3, and FIG. 3B, FIG. 4B, FIG. 5A and FIG. 5B each shows aresult of Gaussian filtration of two-dimensional data.

FIG. 2 shows a result of Gaussian filtration of two-dimensional data inwhich a sinusoidal wave with the amplitude of 1 mm having the samewavelength as the cut-off wavelength is overlaid on a circle with theradius of 10 mm. It can be understood from FIG. 2 that the amplitude ofthe sinusoidal wave was attenuated to 0.5 mm due to the filtration.Namely, it is understood that the amplitude was reduced by 50%.

FIG. 3A and FIG. 3B are views each showing a result of Gaussianfiltration of two-dimensional data in which Gauss noise with thestandard deviation of 0.1 mm is overlaid on a logarithmic spiral. FIG.3A shows a case where the cut-off wavelength is 0.5 mm, and FIG. 3Bshows a case where the cut-off wavelength is 1.0 mm.

FIG. 4A and FIG. 4B are views each showing a result of Gaussianfiltration of two-dimensional data in which Gauss noise with thestandard deviation of 0.1 mm is overlaid on a folium. FIG. 4A shows acase where the cut-off wavelength is 0.5 mm, and FIG. 4B shows a casewhere the cut-off wavelength is 1.0 mm.

FIG. 5A and FIG. 5B are views each showing a result of Gaussianfiltration of two-dimensional data in which Gauss noise with thestandard deviation of 0.1 mm is overlaid on design data for an air foil.FIG. 5A shows a case where the cut-off wavelength is 0.5 mm, and FIG. 5Bshows a case where the cut-off wavelength is 1.0 mm.

As shown in FIG. 3A and FIG. 3B to FIG. 5A and FIG. 5B, it can beunderstood that the same smooth result as that provided by a splinefilter can be obtained when two-dimensional data is subjected toGaussian filtration.

With the first reference technology as described above, the followingeffects are provided.

(1) Even when measurement data is two-dimensional data, Gaussianregression can be carried out. Therefore, It is possible to subjectmeasurement data obtained by scanning the workpiece 10 to be measuredalong a measurement route on a plain with a prespecified pitch tofiltration. This makes it possible to perform form analysis based on themeasurement data obtained by scanning with a prespecified pitch alongthe measurement route, so that, a changing point in a form can begrasped more accurately as compared to a case where measurement data isobtained by scanning with a prespecified pitch in the x-axial direction.For instance, a changing point from a straight region to a circularregion or a border of a step can be grasped more accurately, so thatprecision of form analysis can be improved.(2) Two-dimensional data can be subjected to Gaussian regressionprocessing, so that filtration can be performed suppressing generationof distortions at edge portions of a measurement area.Reference Variant 1

A reference variant 1 of the reference technology for this invention isdescribed below. The basis configuration of the reference variant 1 isthe same as that in the first reference technology, but the referencevariant 1 is characterized in that the computing section 52 executesfiltration of three-dimensional data (x, y, z) obtained bythree-dimensional measurement.

In the computing section 52 executing filtration of thethree-dimensional data as described above, the filter output (g_(x,k),g_(y,k), g_(z,k)) from the Gaussian regression filter 521 is expressedby the following equations:

$\begin{matrix}\begin{matrix}{g_{x,k} = {\sum\limits_{i = 0}^{n - 1}{x_{i} \cdot s_{ik}^{\prime}}}} \\{g_{y,k} = {\sum\limits_{i = 0}^{n - 1}{y_{i} \cdot s_{ik}^{\prime}}}} \\{g_{z,k} = {\sum\limits_{i = 0}^{n - 1}{z_{i} \cdot s_{ik}^{\prime}}}}\end{matrix} & (5)\end{matrix}$

The Gaussian distribution function is expressed like the equation (4)using the sampling pitch Δ1 along a measurement route and the cut-offwavelength λ′_(c) along the measurement route.

With the reference variant 1, the same advantages as those provided bythe first reference technology are provided. Namely, even whenmeasurement data is three-dimensional data, correct filtration can beexecuted by means of Gaussian regression.

Second Reference Technology

A second reference technology for the present invention is describedbelow.

This second reference technology is characterized in that the robustregression is executed by the robust estimation method using weightingfactor for each measurement data in filtration.

The basic configuration of the second reference technology is the sameas that in the first reference technology, but is different from thelatter in configuration of the computing section 52. FIG. 6 is afunctional block diagram showing the computing section 52.

The computing section 52 includes a robust Gaussian regression filter 53(a signal processing filter), and the robust Gaussian regression filter53 includes an initial setting section 54 for initially executingnot-robust filtration of measurement data, and a robust regressionfilter processing section 55 for executing robust filtration byrepeatedly computing a weighting factor.

The initial setting section 54 executes the Gaussian regression in theinitial state, and has the same configuration as that of the Gaussianregression filter 521 in the first reference technology. This is thesame meaning as that all of weighting factor δ are set to “1” in theinitial setting section 54.

The robust regression filter processing section 55 includes a weightingfactor computing section (a weighting factor calculating section) 56 forcomputing (calculating) a weighting factor δ, and a filtration executingsection for computing an output value (g_(x,k), g_(y,k)) by executingcomputing for filtration, and a convergence determining section (aconvergence judging section) 58 for determining (judging) convergence ofa weighting factor δ.

The weighting factor computing section 56 computes a weighting factorfor measurement data.

In the following descriptions, it is assumed that a distance between themeasurement point (x_(i), y_(i)) and a filter output for the measurementpoint is d_(k) and a filter output (a discreteness point) from thefiltration executing section in m-th time processing is (g^(m) _(x,k)g^(m) _(y,k)).d ^(m) _(k)=√{square root over ((x _(i) −g ^(m) _(x,k))²+(y _(i) −g ^(m)_(y,k))²)}{square root over ((x _(i) −g ^(m) _(x,k))²+(y _(i) −g ^(m)_(y,k))²)}  (6)

In this step, the weighting factor computing section 56 computes aweighting factor δmk with the median data β standardized with thestandard deviation σ and a prespecified constant c and using theadaptive type of biweight method.

$\begin{matrix}{\delta_{k}^{m} = \left\{ \begin{matrix}\left\{ {1 - \left( \frac{d_{k}^{m}}{\beta \cdot c} \right)^{2}} \right\}^{2} & {d_{k}^{m} < {\beta \cdot c}} \\0 & {d_{k}^{m} \geqq {\beta \cdot c}}\end{matrix} \right.} & (7)\end{matrix}$

The median data β standardized by a standard deviation is expressed bythe following equation:

$\begin{matrix}{\beta = {{median}\mspace{14mu}\left\{ \frac{d_{k}^{m}}{\sigma} \right\}}} & (8)\end{matrix}$wherein σ is a standard deviation for dmk.

The constant c is decided by the following equation:

$\begin{matrix}{c = \left\{ \begin{matrix}6 & {{\beta \leqq 5}\mspace{56mu}} \\10 & {\mspace{11mu}{5 < \beta \leqq 100}} \\20 & {{100 < \beta}\mspace{34mu}}\end{matrix} \right.} & (9)\end{matrix}$

The weighting factor computing section 56 computes a weighting factor δthrough the equation (7), and outputs the computed weighting factor δ toa filtration executing section 57 and a convergence determining section58. When it is determined by the convergence determining section 58 thatthe weighting factor δ has not been converged, the weighting factorcomputing section 56 updates the weighting factor δ. Namely theweighting factor computing section 56 updates the weighting factor bycomputing a weighting factor δ^(m+1) _(k) through the equation (7) basedon the filter output (g^(m+1) _(x,k), g^(m+1) _(y,k)) by the processingin the next step in the filtration executing section 57.

The filtration executing section 57 executes the Gaussian regressionusing the weighting factor δ computed in the weighting factor computingsection 56.

Namely the output (g^(m) _(x,k), g^(m) _(y,k)) from the robust Gaussianregression filter using the weighting factor δ^(m−1) _(k) for themeasurement data (x_(i), y_(i)) is expressed by the following equations:

$\begin{matrix}{{g_{x,k}^{m} = {\sum\limits_{i = 0}^{n - 1}\;{x_{i} \cdot \delta_{k}^{m - 1} \cdot s_{ik}^{\prime}}}}{{g_{y,k}^{m} = {{\sum\limits_{i = 0}^{n - 1}\;{{y_{i} \cdot \delta_{k}^{m - 1} \cdot s_{ik}^{\prime}}{\mspace{76mu}\;}k}} = 0}},1,2,{{\cdots\mspace{11mu} n} - 1}}} & (10)\end{matrix}$

The Gaussian distribution function s′_(ik) is similarly expressed by theequation (4) with the sampling pitch Δ1 along the measurement route andthe cut-off frequency λ′_(c) along the measurement route. However, theGaussian distribution function s′_(ik) is standardized by the followingequation:

$\begin{matrix}{s_{ik}^{\prime} = \frac{s_{ik}}{\sum\limits_{i = 0}^{n - 1}{\delta_{k}^{m} \cdot s_{ik}}}} & (11)\end{matrix}$

When convergence of the weighting factor δ^(m) _(k) is instructed basedon determination by the convergence determining section 58, thefiltration executing section 57 executes the filtration through theequation (10) with the converged weighting factor δ^(m) _(k). In thisstep, the filter output (g^(m+1) _(x,k), g^(m+1) _(y,k)) is outputtedfrom the filtration executing section 57 to the display section 6.

When it is determined by the convergence determining section 58 that theweighting factor δ^(m) _(k) has not converged yet, the filtrationexecuting section 57 executes filtration with the weighting factor δ^(m)_(k), and outputs this filter output to the weighting factor computingsection 56.

The convergence determining section 58 determines whether the weightingfactor δ^(m) updated in the weighting factor computing section 56 hasconverged or not. Namely the convergence determining section 58 comparesthe weighting factor δ^(m−1) _(k) decided by the filter output (g^(m−1)_(k)) in the (m−1)th step processing to the weighting factor δ^(m) _(k)decided by the filter output (g^(m) _(k)) in the m-th step processing.The convergence determining section 58 makes, in this comparison, thedetermination concerning convergence of the weighting factor δ^(m) _(k)based on the convergence conditions expressed by the following equation:

$\begin{matrix}{{\sum\limits_{k = 0}^{n - 1}\;{{\delta_{k}^{m} - \delta_{k}^{m - 1}}}} < {0.02{\sum\limits_{k = 0}^{n - 1}\;\delta_{k}^{m}}}} & (12)\end{matrix}$

The convergence determining section 58 outputs a result of determinationconcerning convergence of the weighting factor δ^(m) _(k) to thefiltration executing section 57.

Next the robust Gaussian regression for measurement data in the secondreference technology having the configuration as described above isdescribed below. FIG. 7 shows steps in the robust Gaussian regression.

At first, a surface of a workpiece to be measured is scanned with theprobe 21 keeping the z coordinate at a constant value. In this step,coordinate values of the probe 21 are sampled by the position detectingsection 4 with a prespecified pitch Δ1 along the measurement route, andthe sampled measurement data (x_(i), y_(i)) is sent to the computingsection 52 (ST 1).

In the computing section 52, at first, the Gaussian regression isexecuted by the initial setting section 54, and the output (g⁰ _(x,k) g⁰_(y,k)) is sent to the weighting factor computing section 56 (ST 2).

The weighting factor computing section 56 computes the weighting factorδ⁰ _(k) through the equation (10) (ST 3), and outputs a result ofcomputing to the filtration executing section 57 and to the convergencedetermining section 58 (weighting factor calculation step).

The filtration executing section 57 executes computing through theequation (10) using the weighting factor δ⁰ _(k), and obtains a filteroutput (g¹ _(x), g¹ _(y)) (ST 4, filtration executing step).

On the other hand, the convergence determining section 58 compares theweighting factor δ⁰ _(k) computed in the weighting factor computingsection 56 to “1” as a weight initially set and to the equation (12),and makes determination concerning the convergence (ST 5, convergencedetermination step).

When it is determined by the convergence determining section 58 that theweighting factor δ⁰ _(k) does not satisfy the equation for convergencedetermination (12) (ST 6: NO), the filter output (g¹ _(x), g¹ _(y))obtained in the filtration executing section 57 is outputted to theweighting factor computing section 56.

The weighting factor computing section 56 computes the weighting factorδ¹ _(k) in the processing in the next step (ST 8) (ST 3).

Then the processing in steps ST 3 to ST 6 is repeated by the convergencedetermining section 58 until the weighting factor δ^(m) _(k) converges.

When it is determined by the convergence determining section 58 that theweighting factor δ_(m) ^(k) has converged (ST 6: YES), output offiltration (g^(m+1) _(x,k), g^(m+1) _(y,k)) by the filtration executingsection 57 is displayed on the display section 6.

FIG. 8A and FIG. 8B are views each showing a result of robust Gaussianfiltration of two-dimensional data. FIG. 8A shows a result of robustGaussian filtration of data in which a spike noise is added to a foliumwith the cut-off wavelength of 0.1 mm. FIG. 8B shows a result of therobust Gaussian filtration of the data in which a spike noise is addedto design data for an air foil with the cut-off wavelength of 0.5 mm.

As indicates by the results shown in FIG. 8A and FIG. 8B, with therobust Gaussian filtration, a more smooth result as compared to that bythe Gaussian filtration can be obtained suppressing effects by the spikenoise.

With the second reference technology having the configuration asdescribed above, in addition to the effects (1) and (2) provided by thereference technology as described above, the following effects can beprovided.

(3) The robust Gaussian filtration can be executed to two-dimensionaldata. Therefore, form analysis of the workpiece 10 to be measured can beperformed without being affected by locally discrete particular pointdata.

(4) In computing the weighting factor δ, a difference betweenmeasurement data and an output value by filtration is assessed based ona point-to-point distance d_(k) between the measurement data (x_(i),y_(i)) and the output value (g_(x,k), g_(y,k)) by filtration for themeasurement data. By employing the point-to-point distance d_(k) asdescribed above, operating steps for computing are reduced with theoutput response speed improved. For instance, as compared to a casewhere computing is simply performed based on the shortest range betweenmeasurement data and a curve obtained by filtration, work load forcomputing can be reduced.(5) By employing the equation (12) in determination concerningconvergence, determination is made from a change rate of the weightingfactor δ from step to step in the processing. As convergence of filteroutput can be determined in a region where the change rate of theweighting factor δ is smaller than a prespecified value in thedetermination concerning convergence by the robust estimation method, sothat it is not necessary to repeat an unnecessary loop, so that the timerequired for filtration can be reduced.

Further, determination concerning convergence can also be made based ona median (median) of the residual error (point-to-point distance),sometimes convergence of the median does not reflect that of the entireoutputs. In this case, by executing determination concerning convergencebased on a change rate of all weighting factor δ, a correctly convergedfilter output value can be obtained for all measurement data. As aresult, precision in form analysis can be improved.

FIG. 9A and FIG. 9B are views each showing a comparison between thefiltration using the Beaton-Function based on the ISO and the robustGaussian filtration using the adaptive type of biweight method accordingto the present invention. Computing of a weighting factor using theBeaton-Function based on ISO and the determination concerningconvergence in filtration especially for one-dimensional time-seriesdata are described in non-patent document 1, non-patent document 2, andnon-patent document 3.

As understood from the results shown in FIG. 9A and FIG. 9B, in the casewhere the Beaton-Function based on ISO is employed, effects by the spikenoise is not completely suppressed, but with the reference technologyusing the adaptive type of Biweight method, effects by the spike noiseis suppressed and a robust result can be obtained.

Reference Variant 2

A reference technology for the present invention is described below.

The basic configuration in the reference variant 2 is the same as thatin the second reference technology, but the reference variant 2 ischaracterized in that the computing section 52 executes filtration ofthree-dimensional data (x_(i), y_(i), z_(i)) obtained throughthree-dimensional measurement.

In the computing section 52 executing filtration of thethree-dimensional data as described above, a filter output (g_(x,k),g_(y,k), g_(z,k)) from the robust Gaussian filter is expressed by thefollowing equations:

$\begin{matrix}{{g_{x,k}^{m} = {\sum\limits_{i = 0}^{n - 1}{x_{i} \cdot \delta_{k}^{m - 1} \cdot s_{ik}^{\prime}}}}{g_{y,k}^{m} = {\sum\limits_{i = 0}^{n - 1}{y_{i} \cdot \delta_{k}^{m - 1} \cdot s_{ik}^{\prime}}}}{g_{z,k}^{m} = {\sum\limits_{i = 0}^{n - 1}{z_{i} \cdot \delta_{k}^{m - 1} \cdot s_{ik}^{\prime}}}}} & (13)\end{matrix}$

The weighting factor δ is computed through the equation (7) like in thesecond reference technology. In this step, the distance d^(m) _(k)between the measuring point (x_(k), y_(k), z_(k)) and an filter outputvalue (g^(m) _(x,k), g^(m) _(y,k), g^(m) _(z,k)) for this measuringpoint is expressed by the following equation:d ^(m) _(k)=√{square root over ((x _(i) −g ^(m) _(x,k))²+(y _(i) −g ^(m)_(y,k))²+(z _(i) −g ^(m) _(z,k))²)}{square root over ((x _(i) −g ^(m)_(x,k))²+(y _(i) −g ^(m) _(y,k))²+(z _(i) −g ^(m) _(z,k))²)}{square rootover ((x _(i) −g ^(m) _(x,k))²+(y _(i) −g ^(m) _(y,k))²+(z _(i) −g ^(m)_(z,k))²)}  (14)The Gaussian distribution function s′_(ik) is expressed using thesampling pitch Δ1 along the measurement route and a cut-off wavelengthλ′_(c) along the measurement route with the equation (4) like in thefirst reference technology.

With the reference variant 2 as described above, the same effects asthose provided in the second reference technology are provided. Namely,also when the measurement data is three-dimensional data, the robustGaussian filtration can be executed. Therefore, form analysis for athree-dimensional workpiece to be measured can be performed accurately.

Reference Variant 3

A reference variant 3 of the present invention is described below.

The basic configuration of this reference variant is the same as thosein the first reference technology, second reference technology,reference variant 1, and reference variant 2, but is characterized inthat other distribution function is used in place of the Gaussiandistribution function s_(ik).

Namely, in the reference technologies, the box-type function as shown inFIG. 10 may be employed in place of the Gaussian distribution function.This box-type function is f(1)=1/W in a zone [−W/2, W/2], and in areaexcluding the zone [−W/2, W/2], f(1)=0.

By using the box-type function as described above, it is possible toconstruct a movement average filter.

The distribution is required only to satisfy the following equation forstandardization:∫_(−a) ^(a) f(1)d1=1  (15)

First Embodiment

Next a first embodiment of the signal processor according to the presentinvention is described. The basic configuration of the first embodimentis the same as that in the second reference technology, but the firstembodiment is characterized in that median data β is obtained forupdating the weighting factor δ.

In the second reference technology, a weighting factor is updatedthrough the equations (7) to (9), and convergence of the weightingfactor δ is determined by computing through the equation (12).

In the second reference technology, one median data β is obtained forall of data (Refer to the equation (8)).

However, when one median data β is applied to the entire area, localfluctuation of the data can not be grasped.

In the following descriptions, for instance, a case is assumed in whichdata at a very low noise level is processed. In this case, a median ofthe residual error between the measurement data and a filter output(median data β) is very small. Therefore, when a weight is updatedthrough the equation (7), a number of data with the weight of zero andregarded as abnormal data increases. Especially, distortions aregenerated at both edges in the initial processing, so that thedistortions at the two edges become disadvantageously very large due tothe robust-like processing for updating a weight.

Further when the data includes an abrupt step-like fluctuation as shownin FIG. 12, dullness equivalent to the cut-off wavelength is generateddue to the initial processing. Therefore, when a weight is updatedthrough the equation (7), a weight of data at the step-like changingpoint becomes zero, and as a result, a filter output waveform notaccurately reflecting a form is provided by the robust-like processingfor updating a weight.

To overcome this problem, in the first embodiment, a data area isdivided to a plurality of regions, and a median data β is obtained foreach region.

For instance, measurement data is divided to a plurality of zones (firstzones) with a prespecified pitch L as shown in FIG. 13, and median dataβ is computed for each zone through the equation (8). The weightingfactors δ may be computed for measurement data in each zone through theequation (7).

Alternatively, measurement data is divided to zones (second zones) witha prespecified pitch L as shown in FIG. 14, and median data β iscomputed through the equation (8). Then the weighting factor δ may becomputed through the equation (7) for measurement data within a zone(first zone) with a narrower pitch D as compared to those for which themedian data β is computed. Namely, a wider zone is used for computingthe median data β, and this median data β is applied to a narrower zone.A relation between the zone (with the prespecified pitch L) for whichthe median data β is computed and the zone to which the median data β isapplied is as shown by the bar graph in FIG. 14.

In the descriptions above, it is assumed that the length L of each zoneis identical, but the length is not always required to be identical, andeach zone may have any length required for measurement of data.

With the first embodiment as described above, a weight for data in eachof the zones is decided based on the median data β computed for eachzone, so that the filter output well corresponds to data in each zone.

The prespecified pitch L of a zone for which the median data β iscomputed may be, for instance, the same as the cut-off wavelength.

The processing described above may be applied to filtration forexecuting a robust-like processing using the robust Gaussian filter orother distribution function as described in the reference variant 3.Further the processing may be applied to the robust spline filter.

Further the processing may be applied to any filter other than therobust spline filter and the robust Gaussian filter on the conditionthat the robust filter is used for obtaining a filter output value basedon a weighting factor for measurement data (digital signal values) bythe robust estimation method.

FIG. 15 to FIG. 17 are views each showing a result of robust splinefiltration for updating a weighting factor using the median data βobtained in the first embodiment.

FIG. 16 is an enlarged view showing a joint section when the robustspline filter is applied to a two-dimensional curve with small noises asshown in FIG. 15. It is understood from FIG. 16 that, when the mediandata β computed for each of divided zones, a filter output wellfollowing the input data can be obtained.

FIG. 17 shows a case where the robust spline filter is applied to inputdata including a step-like change. It is understood from FIG. 17 that,when the median data β computed for each of divided zones, a filteroutput well following the input data can be obtained.

The signal processing method using the robust spline filter is describedbelow as a reference technology.

Third Reference Technology

At first a weighted spline filter is described.

For an instance, the spline filter is realized by minimizing a sum ofsquares of residual error with the measurement data as expressed by thefollowing equation:

$\begin{matrix}{\sum\limits_{k = 0}^{n - 1}\left\{ {y_{k} - {s\left( x_{k} \right)}} \right\}^{2}} & (16)\end{matrix}$wherein n indicates a number of data, y_(k) (k=0, 1, . . . n) indicatesmeasurement data, s indicates a spline function, on the condition thatthe spline energy as expressed below

$\begin{matrix}{\int_{a}^{b}{\left\{ \frac{\mathbb{d}^{2}{s(x)}}{\mathbb{d}x^{2}} \right\}^{2}\ {\mathbb{d}x}}} & (17)\end{matrix}$is minimized. Namely the spline filter can be realized, when thecondition as defined by the following equation:

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}\left\{ {y_{k} - {s\left( x_{k} \right)}} \right\}^{2}} + {\lambda{\int_{a}^{b}{\left\{ \frac{\mathbb{d}^{2}{s(x)}}{\mathbb{d}x^{2}} \right\}^{2}\ {\mathbb{d}x}}}}}} & (18)\end{matrix}$is satisfied, by obtaining the spline function s minimizing I(s). In theequation above, λ indicates a Lagrange undetermined multiplier.

Assuming that w_(k) (k=0, 1, . . . , n−1) indicates a weight for aresidual error at each measuring point the following equationcorresponding to the weighed spline filter can be obtained.

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}{w_{k}\left\{ {y_{k} - {s\left( x_{k} \right)}} \right\}^{2}}} + {\lambda{\int_{a}^{b}{\left\{ \frac{\mathbb{d}^{2}{s(x)}}{\mathbb{d}x^{2}} \right\}^{2}\ {\mathbb{d}x}}}}}} & (19)\end{matrix}$

By breaking up the spline function s with a constant pitch, and when itis assumed that the second term is expressed as shown below:

$\begin{matrix}{\alpha{\sum\limits_{k = 0}^{n - 1}{\nabla^{2}{s\left( x_{k} \right)}}}} & (20)\end{matrix}$the following equation can be obtained.

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}{w_{k}\left\{ {y_{k} - {s\left( x_{k} \right)}} \right\}^{2}}} + {\alpha{\sum\limits_{k = 0}^{n - 1}{\nabla^{2}{s\left( x_{k} \right)}}}}}} & (21)\end{matrix}$Herein it is assumed that the conditions expressed by the followingequation is satisfied:∇² s(x _(k))=s(x _(k+1))−2s(x _(k))+s(x _(k−1))  (22)Therefore, the value sk of broken-up spline minimizing I(s) satisfiesthe following equation:

$\begin{matrix}{{\frac{\partial{I\left( {S_{0},S_{1},{\cdots\mspace{11mu} S_{n - 1}}} \right)}}{\partial s_{k}} = {{0\mspace{25mu} k} = 0}},1,\cdots\mspace{11mu},{n - 1}} & (23)\end{matrix}$

Then the weighted spline filter minimizing I(s) in the equation (21) isdefined.

Assuming a matrix expression for a weighted spline filter for non-cyclicmeasurement data, and the boundary condition for the non-cyclicmeasurement data is expressed by the following equations:∇² s(x ₀)=0 ∇² s(x _(n−1))=0  (24)the weighted spline filter is expressed as follows:

$\begin{matrix}{{{{\frac{\partial I}{\partial s_{0}} = {{{- 2}{w_{0}\left( {y_{0} - s_{0}} \right)}} + {2{\alpha\left( {s_{2} - {2s_{1}} + s_{0}} \right)}}}}\frac{\partial I}{\partial s_{1}} = {{{- 2}{w_{1}\left( {y_{1} - s_{1}} \right)}} + {2{\alpha\left( {s_{3} - {4s_{2}} + {5s_{1}} + {2s_{0}}} \right)}}}}{{\frac{\partial I}{\partial s_{k}} = {{{{- 2}{w_{k}\left( {y_{k} - s_{k}} \right)}} + {2{\alpha\left( {s_{k + 2} - {4s_{k + 1}} + {6s_{k}} - {4s_{k - 1}} + s_{k - 2}} \right)}k}} = 2}},3,\cdots\mspace{11mu},{n - 3}}\frac{\partial I}{\partial s_{n - 2}} = {{{- 2}{w_{n - 2}\left( {y_{n - 2} - s_{n - 2}} \right)}} + {2{\alpha\left( {s_{n - 4} - {4s_{n - 3}} + {5s_{n - 2}} - {2s_{n - 1}}} \right)}}}}{\frac{\partial I}{\partial s_{n - 1}} = {{{- 2}{w_{n - 1}\left( {y_{n - 1} - s_{n - 1}} \right)}} + {2{\alpha\left( {s_{n - 3} - {2s_{n - 2}} + s_{n - 1}} \right)}}}}} & (25)\end{matrix}$and when the following substitution is employed:

$\begin{matrix}{Q = \begin{pmatrix}1 & {- 2} & 1 & \; & \; & \; & \; \\{- 2} & 5 & {- 4} & 1 & \; & \; & \; \\1 & {- 4} & 6 & {- 4} & 1 & \; & \; \\\; & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \; \\\; & \; & 1 & {- 4} & 6 & {- 4} & 1 \\\; & \; & \; & 1 & {- 4} & 5 & {- 2} \\\; & \; & \; & \; & 1 & {- 2} & 1\end{pmatrix}} & (26)\end{matrix}$a matrix expression of the weighted spline filter for non-cyclic data isgiven by the following equation:(W+αQ)S=WY  (27)wherein W, S and Y satisfies the following expressions respectively:

$\begin{matrix}{W = {{\begin{pmatrix}W_{0} & \; & \; & \; & \; & \mspace{11mu} \\\; & W_{1} & \; & \; & \; & \; \\\; & \; & ⋰ & \; & \; & \; \\\; & \; & \; & W_{n - 3} & \; & \; \\\; & \; & \; & \; & W_{n - 2} & \; \\\; & \mspace{11mu} & \; & \; & \; & W_{n - 1}\end{pmatrix}S} = {{\begin{pmatrix}s_{0} \\s_{1} \\\vdots \\s_{n - 3} \\s_{n - 2} \\s_{n - 1}\end{pmatrix}Y} = \begin{pmatrix}y_{0} \\y_{1} \\\vdots \\y_{n - 3} \\y_{n - 2} \\y_{n - 1}\end{pmatrix}}}} & (28)\end{matrix}$

As for the matrix expression of a weighted spline filter for cyclicmeasurement data, assuming that the cyclic boundary condition isexpressed by the following equations:s _(k+n) =S _(k) k=0, 1, . . . ,n−1  (29)the following conditions is satisfied,∂I/∂S _(k)=−2w _(k)(y _(k) −s _(k))+2α(s _(k+2)) −4s _(k+1)+6s _(k)−4s_(k−1) +s _(k−2))k=0, 1, . . . ,n−1  (30)and therefore assuming that the following condition is satisfied:

$\begin{matrix}{\overset{\sim}{Q} = \begin{pmatrix}6 & {- 4} & 1 & \; & \; & 1 & {- 4} \\{- 4} & 6 & {- 4} & 1 & \; & \; & 1 \\1 & {- 4} & 6 & {- 4} & 1 & \; & \; \\\; & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \; \\\mspace{11mu} & \; & 1 & {- 4} & 6 & {- 4} & 1 \\1 & \mspace{25mu} & \; & 1 & {- 4} & 6 & {- 4} \\{- 4} & 1 & \; & \; & 1 & {- 4} & 6\end{pmatrix}} & (31)\end{matrix}$the matrix expression for a weighted spline filter for cyclic is givenby the following equation:(W+α{tilde over (Q)})S=WY  (32)

The amplitude characteristic and phase characteristic of the splinefilter are discussed below.

When the following equation with the weight W of 1 (unit matrix):y _(k) =s _(k)+α(s _(k+2)−4s _(k+1)+6s _(k)−4s _(k−1) +s _(k−2))k=0, 1, . . . ,n−1  (33)is subjected to z-transformation taking into consideration that z⁻¹indicates a delay for Δx, the following equation is given:y _(k) =s _(k)+α(z ⁻²−4z ⁻¹+6−4z+z ²)s _(k)  (34)

The transfer function (Hz) for a spline filter is given by the followingequation:

$\begin{matrix}\begin{matrix}{{H(z)} = \frac{s_{k}}{y_{k}}} \\{= \frac{1}{1 + {\alpha\left( {z^{- 2} - {4z^{- 1}} + 6 - {4z} + z^{2}} \right)}}}\end{matrix} & (35)\end{matrix}$To examine the amplitude characteristic and phase characteristic,assuming the following equation:z=e ^(jωΔx)  (36)the condition as expressed by the following equation is satisfied:

$\begin{matrix}{{H(\omega)} = \frac{1}{1 + {\alpha\left( {{\mathbb{e}}^{{- 2}j\;\omega\;\Delta\; x} - {4{\mathbb{e}}^{{- j}\;\omega\;\Delta\; x}} + 6 - {4{\mathbb{e}}^{j\;\omega\;\Delta\; x}} + {\mathbb{e}}^{j\;\omega\;\Delta\; x}} \right)}}} & (37)\end{matrix}$

Herein the following condition is satisfied:e ^(−jα) +e ^(jα)=2 cos α  (38)the following equation is obtained:

$\begin{matrix}{{{\mathbb{e}}^{{- 2}j\;\omega\;\Delta\; x} - {4{\mathbb{e}}^{{- j}\;\omega\;\Delta\; x}} + 6 - {4{\mathbb{e}}^{j\;\omega\;\Delta\; x}} + {\mathbb{e}}^{2j\;\omega\;\Delta\; x}}\begin{matrix}{= {{2{\cos\left( {2\;{\omega\Delta}\; x} \right)}} - {8{\cos\left( {{\omega\Delta}\; x} \right)}} + 6}} \\{= {2 - {4{\sin^{2}\left( \;{{\omega\Delta}\; x} \right)}} - {8\cos\left( {{\omega\Delta}\; x} \right)} + 6}} \\{= {{{- 16}{\sin^{2}\left( \frac{{\omega\Delta}\; x}{2} \right)}{\cos^{2}\left( \frac{{\omega\Delta}\; x}{2} \right)}} - 8 + {16{\sin^{2}\left( \frac{{\omega\Delta}\; x}{2} \right)}} + 8}} \\{= {16{\sin^{4}\left( \frac{{\omega\Delta}\; x}{2} \right)}}}\end{matrix}} & (39)\end{matrix}$and the amplitude characteristic is expressed by the following equation:

$\begin{matrix}{{{H(\omega)}} = \frac{1}{{1 + {16\alpha\mspace{14mu}\sin^{4}\;\left( \frac{{\omega\Delta}\; x}{2} \right)}}\mspace{11mu}}} & (40)\end{matrix}$

On the other hand, the phase characteristic is expressed by thefollowing equation:Arg·H(ω)=0  (41)and therefore it can be understood that the spline filter is aphase-compensation filter.

For instance, when a filter with 50% attenuation is realized with thecut-off frequency ω=ωw, the phase characteristic is required only tosatisfy the following condition:

$\begin{matrix}{{H\left( \omega_{c} \right)} = \frac{1}{2}} & (42)\end{matrix}$and the constant α is given by the following equation:

$\begin{matrix}{\alpha = \frac{1}{16{\sin^{4}\left( \frac{\omega_{c}\Delta\; x}{2} \right)}}} & (43)\end{matrix}$

When the cut-off frequency ω is ω_(c), the transfer characteristic(amplitude characteristic, phase characteristic) of the filter with 50%attenuation is as shown in FIG. 22.

Next, a solution for the weighted spline filter defined as describedabove is discussed below.

The matrix expression for a weighted spline filter is expressed by thefollowing equation:(W+αQ)S=WY  (44)and the matrix in the left side as expressed by the following equation:M=W+αQ  (45)is a symmetric matrix.

When M is broken down to the low triangle matrix L and a diagonal matrixD by the amended Cholesky method (a matrix can very efficiently bebroken down by making use of the fact that M is a sparse matrix toobtain the following equation:M=LDL^(T)  (46)and the weighted spline filter is expressed by the following equation:LDL^(T)S=WY  (47)

Assuming therein as follows:DL^(T)S=X  (48)the following equation is given:LX=WY  (49)

As L is a lower triangle matrix, X can easily be obtained. Furtherbecause of the relation as expressed by the following equation:L ^(T) S=D ⁻¹ X  (50)S can easily be obtained from X obtained as described above.

In the practical application, the condition as expressed by thefollowing equation may occur:W_(k) ^(m)=0  (51)and in that case the matrix M may be a singular one.

Therefore, it is preferred to solve the equation by the singular valuedecomposition method, but when the singular value decomposition methodis employed, a storage device with a large memory capacity and anextremely long period of time for computing are required. However, inapplication to actual measurement data, the matrix M rarely becomessingular, and in the state where the matrix M is singular, thepossibility may be suspected that the measurement data itself containsome problems. Therefore, in the present invention, by applying theamended Choleskey method by Gill-Murray capable of giving a solutioneven when the matrix M is singular, the two workpieces of securing thecomputing efficiency and countermeasures against a singular matrix aresimultaneously achieved.

As described above, as a weighted spline filter supported by the solvingmethod as described above was derived, the robust spline filter can berealized by repeating computing until the condition for convergence issatisfied by updating a weight W.

FIG. 18 is a flow chart showing the first processing sequence, and FIG.19 is a functional block diagram showing a device for executing therobust spline processing. When executing this processing, at first, ameasurement data input step of inputting measurement data and aselection step of selecting a weighted spline filter equation (ST 103)are executed.

In the measurement data input step, a step ST 101 for receivingmeasurement data from an input unit 101 such as an measuring instrumentand storing the data in a storage device 102 such as a computer and astep ST 102 of deleting singular point data locally isolated with asingular point data deleting unit 103 are executed.

In the following descriptions, it is assumed that the measurement datais one-dimensional time series data obtained by a measuring instrument.Namely, the case is assumed, for instance, in which, when a probe of,for instance, a surface roughness measuring instrument is moved in onedirection (x-axial direction), roughness data y is obtained with theprespecified pitch in the x-axial direction. The determination as towhether data is singular ones or not can easily be made by determiningwhether the difference from minimum square curve of the measurement datais not less than a prespecified value and not more than a prespecifiedzone width or not.

Then in the selection step ST 103, determination is made by adetermining unit 104 as to whether the measurement data is non-cyclic orcyclic, and a weighted spline filter equation is selected based on aresult of this determination. More specifically, which of the equation(27) and equation (32) is to be used is determined depending on a resultof determination as to whether the measurement data is non-cyclic orcyclic.

The processing for initialization (ST 104) is executed, and in thisstep, an initial value S0 of an output value by the spline filtration isobtained assuming that W is equal to I (unit matrix) (non-robust splinefiltration).

Then the weight W^(m) is adjusted and decided with a weight adjuster 105from the measurement data Y and S^(m) (m indicates a repeated step)employing the method described hereinafter (ST 105).

Then a spline filter output S^(m+1) is obtained from the weighted splinefilter as expressed by the following equation:(W ^(m) +αQ)S ^(m+1) =W ^(m) Y  (52)in a spline filter output computing unit 106 (ST 106).

Herein determination concerning convergence of a weight as describedbelow (ST 107) is executed by a convergence determining unit 151, andwhen it is determined that the conditions for convergence have not beenestablished, m is updated (m=m+1) (ST 110), and again the weight W^(m)is adjusted (ST 105).

When it is determined that the conditions for convergence have beenestablished (ST 107, YES), the repetitive processing is terminated toobtain an output value S^(m) from the robust spline filter (ST 108), anda spline curve is outputted to an output unit 107.

In the processing described above, as a method of adjusting and decidingthe weight W^(m) (ST 105), the adaptive type of biweight method isemployed to decide the weight W^(m) as expressed by the followingequation:

$\begin{matrix}{W_{k}^{m} = \left\{ \begin{matrix}\left\lbrack {1 - \left( \frac{y_{k} - s_{k}^{m}}{\beta \cdot c} \right)^{2}} \right\rbrack^{2} & {{{y_{k} - s_{k}^{m}}} < {\beta \cdot c}} \\0 & {{{y_{k} - s_{k}^{m}}} \geq {\beta \cdot c}}\end{matrix} \right.} & (53)\end{matrix}$wherein β and C are expressed by the following equations:

$\begin{matrix}{\beta = {{median}\left\{ {\frac{y_{k} - s_{k}^{m}}{\sigma}} \right\}}} & (54) \\{c = \left\{ \begin{matrix}{6} & {\beta \leq 5} \\10 & {5 < \beta \leq 100} \\{20} & {100 < \beta}\end{matrix} \right.} & (55)\end{matrix}$wherein σ indicates a standard deviation of the residual error.

As for the conditions for convergence in ST 107, the repetitiveprocessing is terminated at a point of time when change of the weightbecomes sufficiently small and the equation shown below is satisfied:

$\begin{matrix}{{\sum\limits_{k = 0}^{n - 1}{{w_{k}^{m} - w_{k}^{m - 1}}}} < {0.02 \cdot {\sum\limits_{k = 0}^{n - 1}w_{k}^{m}}}} & (56)\end{matrix}$

FIG. 20A and FIG. 20B each shows an example in which the signalprocessing method by robust spline filtration based on this thirdreference technology is executed to one-dimensional time-series data.These figures show the signal processing method for measurement datawith spike noise added thereto, and a spline curve as a result ofordinary spline filtration and a spline curve as a result of robustspline filtration according to the present invention are shown in thefigures respectively. As clearly understood from the figures, the resultof spline filtration is affected by the spike noise, while the result ofrobust spline filtration shows a spline curve reflecting the actualform. Further as understood from FIG. 20A, the result of robust splinefiltration is excellent in the following capability for a form havingmild wavinesss.

With the method as described above, the following advantages can beexpected.

As a spline filter can easily be changed to a robust spline filter,deformation at a measurement start point and that at a measurement endpoint can be prevented, and as a form included in measurement data canbe extracted without being affected by the following capability towaviness components each with a long cycle contained in the measurementdata nor by noise components, filtration can be executed with highcapability for following a form of a workpiece to be measured, so thatreliability of the measurement data is further improved.

As locally isolated data at a singular point contained in measurementdata can be deleted, so that the reliability of the robust splinefiltration is further improved.

The larger the difference of measurement data from a spline curvecomputed through the weighted spline filter equation is, the smaller theweight is, so that the robust spline filtration not affected by abnormaldata contained in measurement data can be executed.

When a change of a weight drops to a prespecified value or below in therepetitive loop processing, it can be determined that the weight hasconverged, so that unnecessary repetitive loop processing can beprevented, and a period of time required for the robust splinefiltration can be shortened.

Fourth Reference Technology

The second processing sequence for realizing the robust spline filter isdescribed below. The processing sequence is the same as the firstprocessing sequence, but is different from the latter in the equationsused for computing.

The equation for the weighted spline filter is prepared by deforming theequation:(W+αQ)S=WY  (57)to obtain the following equation:(I+αQ)S=WY+(I−W)S  (58)

In the repetitive step m, the following equation is used:(I+αQ)S ^(m+1) =W ^(m) Y+(I−W ^(m))S ^(m)  (59)As a feature of this second processing sequence, in addition to theadvantages provided by the first processing sequence, the followingadvantages can be expected. Namely, a coefficient matrix in the leftside expressed below:I+αQ  (60)has always the same value in the repetitive steps, so that a totalperiod of time required for the robust spline filtration may beshortened.Fifth Reference Technology

As a fifth reference technology according to the signal processingmethod of the present invention, the signal processing method formeasurement data which is two-dimensional data obtained by means oftwo-dimensional measurement is described below. The measurement data astwo-dimensional data as used herein indicates, for instance, (x, y)coordinate values or the like obtained by using, for instance, acoordinate measuring machine for measuring a contour curved surface of aworkpiece to be measured with a prespecified pitch and keeping the zcoordinate unchanged. The measurement data also includes data obtained,for instance, when a figure drawn on a plain is read with a scanner.Namely in the first reference technology, a workpiece for measurement isonly y coordinate, but in the fifth reference technology, workpieces formeasurement are x coordinate and y coordinate.

The basic configuration of the fifth reference technology is the same asthat of the third reference technology, and is characterized in theequation corresponding to the equation (21) which is an initial equationfor obtaining a spline curve s.

In the fifth reference technology, measurement data (x_(k), y_(k)) and aspline curve providing a minimum sum of squares of distances in thex-axial direction and y-axial direction with a point (s_(x) (x_(k),y_(k)), s_(y) (x_(k), y_(k))) on a spline curve corresponding to themeasurement data are obtained under the condition that the spline energyis minimized. Namely, the spline function s minimizing I(s) expressed bythe following equation is obtained under the additional conditiondescribed above:

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}{w_{k}\left\lbrack {\left\{ {x_{k} - {s_{x}\left( {x_{k},y_{k}} \right)}} \right\}^{2} + \left\lbrack {y_{k} - {s_{y}\left( {x_{k},y_{k}} \right)}} \right\}^{2}} \right\rbrack}} + {\alpha{\sum\limits_{k = 0}^{n - 1}\left\{ {{\nabla_{x}^{2}{s\left( {x_{k},y_{k}} \right)}} + {\nabla_{y}^{2}{s\left( {x_{k},y_{k}} \right)}}} \right\}}}}} & (61)\end{matrix}$wherein, in the second term in the right side, the Laplace secondaryapproximation is expressed as shown below:∇_(x) ² s(x _(k) ,y _(k))=s _(x)(x _(k+1))−2s _(x)(x _(k))+s _(x)(x_(k−1))∇_(y) ² s(x _(k) ,y _(k))=s _(y)(y _(k+1))−2s _(y)(y _(k))+s _(y)(y_(k−1))  (62)

Then the weighted spline filtration described in the third referencetechnology is executed for x component and y component respectively(Refer to the equation (52)).

Herein the constant α is given based on the sampling pitch Δ1 and thecut-off wavelength λ′_(c) each along a measurement route by computingwith the following equation:

$\begin{matrix}{\alpha = \frac{1}{16{\sin^{4}\left( \frac{{\pi \cdot \Delta}\; I}{\lambda_{C}^{\prime}} \right)}}} & (63)\end{matrix}$

With the operations described above, a spline filter for deriving aspline curve for each zone in the two-dimensional data is realized.

Further in the robust spline filter repeating the processing until theequation (56) for convergence condition is satisfied by updating theweight W, (y^(k)−s^(k) _(m)) in the equation (53) is used as thepoint-to-point distance in the following equation. Namely (y^(k)−s^(k)_(m)) in the equation (53) is used as a distance between the measurementdata (x_(k), y_(k)) and a point (s_(x)(x_(k), y_(k)), s_(y)(x_(k),y_(k))) on a spline curve s corresponding to the measurement data(x_(k), y_(k)):d _(k)=√{square root over ({x _(k) −s _(x)(x _(k) ,y _(k))}² +{y _(k) −s_(y)(x_(k), y_(k))}²)}{square root over ({x _(k) −s _(x)(x _(k) ,y_(k))}² +{y _(k) −s _(y)(x_(k), y_(k))}²)}  (64)

Convergence of the weight W computed through the equation (53) derivedfrom the equation (64) is used for determination thereof in the equation(56). When the weight W has converged, a spline curve corresponding tothe measurement data is obtained from the output value S^(m) (splinefunction). This spline curve is outputted to the output unit.

FIG. 21A shows a comparison between a result of spline processing ofinput data in which spike noise is added to a folium line and a resultof robust spline processing of the same measurement data. It isunderstood from FIG. 21A that the result of simple spline processing isaffected by the spike noise, but that the result of the robust splineprocessing is robust without being affected by the spike noise. FIG. 21Bshows a comparison between a result of spline processing of input datain which spike noise is added to an air foil and a result of the robustspline processing of the same measurement data, and the result is thesame as that shown in FIG. 21A.

In this fifth reference technology, in addition to the advantagesprovided in the third and fourth reference technologies described above,the following advantages are provided.

When the measurement data is two-dimensional data in an orthogonalcoordinate system, a difference of measurement data from a spline curveis decided by a sum of squares of a component in each axial direction(for instance, x-axial component, y-axial component), so that thedifference can easily be computed. Therefore a weight for eachmeasurement data can easily be decided.

Even when the measurement data is two-dimensional data, a spline filteroutput can be obtained based on a result of computing with a weightedspline filter for each component in each axial direction (for instance,x-axial component, y-axial component), so that computing can easily becarried out even for a complicated curve, and therefore a period of timerequired for robust spline filtration for measurement data can beshortened.

Even when a workpiece for measurement is measured by scanning in thetwo-dimensional plain to obtain two-dimensional data and thetwo-dimensional data is inputted as measurement data, as the measurementdata is inputted with a prespecified space along a measurement route, aform changing point (such as, for instance, a changing point from astraight portion to a circular portion or a boundary point of a step)can be grasped more accurately as compared to a case in whichmeasurement data is inputted, for instance, with a prespecified space inthe x-axial direction. Namely, erroneous determination concerning a formcan be prevented, which enables input of more reliable measurement data.

Sixth Reference Technology

Next a signal processing method for measurement data which isthree-dimensional data obtained by three-dimensional measurement isdescribed as a sixth reference technology according to the signalprocessing method of the present invention. In the followingdescriptions, the three-dimensional measurement data means (x, y, z)coordinate values and the like obtained by measuring a surface of aworkpiece for measurement with a prespecified pitch using, for instance,a coordinate measuring machine. Namely in the third referencetechnology, a workpiece of the processing is only y coordinate, but inthe sixth reference technology, three factors of x coordinate, ycoordinate, and z coordinate are processed.

Also the basic configuration of the sixth reference technology is thesame as that of the third reference technology, but is characterized inthe equation corresponding to the equation (21) which is a startingequation for obtaining a spline curve s.

In the sixth reference technology, a spline curve minimizing a sum ofsquares of distances in the x-, y-, and z-axial directions between themeasurement data (x_(y), y_(k), z_(k)) and a point (s_(x)(x_(k), y_(k),z_(k)), s_(y)(x_(k), y_(k), z_(k)), s_(z)(x_(k), y_(k), z_(k))) on aspline curve s corresponding to the measurement data (x_(y), y_(k),z_(k)) is obtained under the condition that the spline energy isminimized. Namely, the spline function s minimizing the I(s) expressedby the following equation is obtained under the additional conditiondescribed above:

$\begin{matrix}{{I(s)} = {{\sum\limits_{k = 0}^{n - 1}\left\lbrack {\left\{ {x_{k} - {s_{x}\left( {x_{k},y_{k},z_{k}} \right)}} \right\}^{2} + \left\{ {y_{k} - {s_{y}\left( {x_{k},y_{k},z_{k}} \right)}} \right\}^{2} + \left\{ {z_{k} - {s_{z}\left( {x_{k},y_{k},z_{k}} \right)}} \right\}^{2}} \right\rbrack} + {\alpha{\sum\limits_{k = 0}^{n - 1}\left\{ {{\nabla_{x}^{2}{s\left( {x_{k},y_{k},z_{k}} \right)}} + {\nabla_{y}^{2}{s\left( {x_{k},y_{k},z_{k}} \right)}} + {\nabla_{z}^{2}{s\left( {x_{k},y_{k},z_{k}} \right)}}} \right\}}}}} & (65)\end{matrix}$wherein, in the second term in the right side, a Laplace secondaryapproximation is expressed like in the fifth reference technology.

Then the weighted spline filtration as described in the third referencetechnology is executed for each of the x component, y component, and zcomponent respectively (Refer to the equation (52)). The constant α isdefined with the sampling pitch Δ1 and the cut-off wavelength λ′_(c)along a measurement route in a three-dimensional space through theequation (63).

Then a spline filter for deriving a spline curve in each zone forthree-dimensional measurement data is realized.

Further in the robust spline filter repeating the processing until theequation (56) for convergence condition is satisfied by updating theweight W, (y^(k)−s^(k) _(m)) in the equation (53) is used as apoint-to-point distance given in the following equation. Namely(y^(k)−s^(k) _(m)) is used as a distance between measurement data(x_(y), y_(k), z_(k)) and a point (s_(x)(x_(k), y_(k), z_(k)),s_(y)(x_(k), y_(k), z_(k)), s_(z)(x_(k), y_(k), z_(k))) on a splinecurves corresponding to the measurement data (x_(y),y_(k), z_(k)):d _(k)=√{square root over ({x _(k) −s _(x)(x _(k) ,y _(k) ,z _(k))}² +{y_(k) −s _(y)(x _(k) ,y _(k) ,z _(k))}² +{z _(k) −s _(z)(x _(k) ,y _(k),z _(k))}²)}{square root over ({x _(k) −s _(x)(x _(k) ,y _(k) ,z _(k))}²+{y _(k) −s _(y)(x _(k) ,y _(k) ,z _(k))}² +{z _(k) −s _(z)(x _(k) ,y_(k) ,z _(k))}²)}{square root over ({x _(k) −s _(x)(x _(k) ,y _(k) ,z_(k))}² +{y _(k) −s _(y)(x _(k) ,y _(k) ,z _(k))}² +{z _(k) −s _(z)(x_(k) ,y _(k) ,z _(k))}²)}  (66)

Convergence of the weight W computed through the equation (53) derivedfrom the equation (66) is determined with the equation (56). When theweight W has converged, a spline curve corresponding to measurement datais obtained from the output value S^(m) (spline function). This splinecurve is outputted to the output unit.

In the sixth reference technology described above, in addition to theadvantages provided in the third and fourth reference technologies, thefollowing advantages are provided.

The advantages provided by the fifth reference technology can appliedeven to three-dimensional data. Therefore, even when the measurementdata is three-dimensional data, workload for computing can be reducedwithout increase in a period of time required for robust splinefiltration.

Reference Variant 4

A reference variant 4 of the signal processing method according to thepresent invention is described below. In the third reference technology,a case is described in which a spline curve at a point of time when itis determined that convergence has occurred is outputted as it is as aresult of signal processing, but in this reference variant 4, the splinecurve is again computed and then a result of computing is outputted as aresult of signal processing.

FIG. 23 shows a variant of the spline curve output (ST 109) in FIG. 18.

Herein, at first the fetched output value Sm is inputted (ST 191). Then,whether re-computing is required or not is determined (ST 192). Theoperator is required at the time point only to input YES when a resultof signal is to be obtained with high precision, and NO when it isdetermined that a result was obtained with sufficient precision.Alternatively, the operator may previously specify whether a result ofsignal processing should be provided with high precision or not.

When it is determined that re-computing is not required (NO), a splinecurve corresponding to the output value Sm is outputted by an outputunit 7. When it is required that re-computing is required (YES), aweight surpassing a prespecified value is updated to 1 (ST 193). Namely,measurement data having a weight larger than the prespecified value isregarded as effective data, and a contribution degree thereof to splinecomputing is regarded as 100%.

Then an output is obtained by executing weighted spline filtration basedon the updated weight (ST 194). The spline curve obtained in this stepis outputted as a result of signal processing to the output unit 7 (ST195).

As the reference variant 4 can be carried out in the third to sixthreference technologies, in addition to the advantages as describedabove, the following advantages are provided.

When it is determined in the convergence determination step that weightconvergence has occurred, if the weight is over a prespecified value,the weight is updated to 1 to obtain a spline filter output again, andthe result can be outputted as a result of signal processing. Namely,the weight adjustment step and the spline filter output computing stepare repeated, and when it is determined that the weight has converged,the measurement data at points where weights are over the prespecifiedvalue are regarded as effective data, and the weights are updated to 1,and thus as the spline filter output can be obtained again, the robustspline filtration for measurement data can be executed more accurately.Then the result is outputted as a result of signal processing, namely asa spline curve with a sufficiently small error from the original formcomponent included in the measurement data, the robust spline filtrationwell following an actual form can be executed.

Other Aspects of the Present Invention

A signal processing method for executing filtration of measurement dataat a specified dimension according to one aspect of the presentinvention includes a measurement data input step of inputting themeasurement data along a measurement route; a selection step ofselecting an equation for weighted spline filtration according to a typeof the measurement data; an initialization step of giving a weight forthe measurement data as a unit matrix to obtain an initial value of aspline filter output; an adjustment step of adjusting and deciding aweight for the measurement data; a spine filter output computing step ofobtaining a spline filter output by using the weight decided in theweight adjustment step; a convergence determination step of determiningwhether the weight has converged or not; and an output step ofoutputting a result of signal processing based on the spline filteroutput, and in this aspect, preferably the robust spline filtrationshould be executed to the measurement data, when it is determined in theconvergence determination step that the weight has not converged yet, byupdating the weight and repeating the weight adjustment step and thespline filter output computing step.

With this aspect, an equation for weighted spline filtration isselected, and a spline curve as a spline filter output is repeatedlycomputed by successively updating the weight based on the selectedequation for spline filtration, so that the robust spline filtration ofthe measurement data can easily be executed assuming the spline curve atthe time point of convergence of the weight as a filter output which isa result of signal processing. Because of the features as describedabove, the end-effects at a start point of measurement data and also atan end point thereof can be prevented, and a form included inmeasurement data can be extracted without being affected by thefollowing capability to waviness components each having a long cycle norby noise components contained in the measurement data. As a result,filtration well reflecting the actual form can be performed, so that thereliability of measurement data is further improved.

The measurement data at a prespecified dimension as used hereinindicates measurement data such as one-dimensional time series data(such as, for instance, data obtained by measuring displacements in they-axial direction at positions with a prespecified space therebetween inthe x-axial direction on an orthogonal x and y coordinate plain),two-dimensional data (free curve data on an X-Y plain defined by x-axisand y axis crossing each other at right angles), three-dimensional data(a free space curve data in a x-y-z space defined by x axis, y axis, andz axis crossing at right angles), or a polar coordinates defined by aradius and an angle.

Input of measurement data along a measurement route as used hereinindicates inputting measurement data in a prespecified scanningdirection or inputting measurement data obtained by measuring a surfaceof a workpiece to be measured.

Further in this aspect, the weight decided in the weight adjustment stepshould preferably as small as possible when a difference of themeasurement data from the spline curve computed through the equation forweighted spline filter is large.

With this aspect, the larger the difference of the measurement data fromthe spline curve computed through the equation for weighted splinefilter is, the smaller the weight is, so that the robust splinefiltration can be carried out without being affected by abnormal dataincluded in the measurement data. Namely, a spline curve is repeatedlyobtained by making smaller the weight for the measurement data away fromthe spline curve and also making larger the weight for the measurementdata close to the spline curve. With the operations, the spline curvegradually gets closer to the actual form components (such as true valuesfor an actual form of the workpiece to be measured or the like). Thefinal spline curve obtained the time point when it is determined thatthe weight has converged is obtained as a form component with asufficiently small error from the actual form component. As a result,the extremely excellent robust spline filtration can be carried out.

Further in this aspect, the prespecified dimension of the measurementdata preferably includes two- or three-dimensional component, and thedifference of the measurement data is decided by a sum of squares ofeach component.

Further with this aspect, when the measurement data is two-dimensionaldata or three-dimensional data in an orthogonal coordinate system, adifference of the measurement data from the spline curve is decidedbased on a sum of squares of each of the component in each axialdirection (such as x-axial component, y-axial component, z-axialcomponent, and the like), so that the difference can easily be computed.For the features as described above, a weight for each measurement datacan easily be decided.

In the convergence determination step according to this aspect, it ispreferable to determine that the weight has converged when a change ofthe weight decided in the weight adjustment step drops to a prespecifiedvalue or below.

In this aspect of the present invention, it can be determined, when achange of a weight in the repetitive loop processing had dropped to aprespecified value or below, that the weight has converged, so thatincrease of processing time due to unnecessary repetitive loop can beprevented, and the time required for robust spline filtration can beshortened. Further when a change of the weight dropped to a prespecifiedvalue or below, it can be determined that an error of the spline curvefrom the actual form component included in the measurement data hasbecome sufficiently small, so that the robust spline filtration can beexecuted in the extremely excellent state.

In this aspect of the present invention, the output step preferablyincludes an updating step of updating, when a weight for the measurementdata is over a prespecified value, the weight to 1; a spline filterre-output computing step of obtaining a spline filter output based onthe updated weight; and a signal processing result output step ofoutputting the spline filter output in the spline filter re-outputcomputing step as a result of signal processing.

In this aspect of the present invention, when it is determined in theconvergence determination step that the weight has converged, if theweight is over a prespecified value, the weight is updated to 1 toobtain a spline filter output again, and the result can be outputted asa result of signal processing. Namely, at a point of time when it isdetermined, after the weight adjustment step and spline filter outputcomputing step are repeated several times, that the weight hasconverged, the measurement data with the weight over a prespecifiedpoint is regarded as effective data with the weight updated to 1, andthen a spline filter output is obtained again. With the operationsabove, the robust spline filtration for measurement data can beperformed more accurately. Then the result is outputted as a result ofsignal processing, so that it is possible to obtain a spline curve witha sufficiently small error against the actual form component included inthe measurement data. As a result, the robust spline filtration wellreflecting the actual form can be performed.

Further in this aspect of the present invention, the measurement datapreferably includes components at two- or higher dimension in anorthogonal coordinate system, and when the spline filter output is to beobtained, the spline filter output is preferably obtained based on aresult of weighted spline filtration for each of the components.

In this aspect of the present invention, even when measurement data istwo-dimensional data or three-dimensional data, a spline filter outputcan be obtained based on a result of a weighted spline filtration foreach component in each axial direction (such as x-axial component,y-axial component), and therefore even computing for a complicated curvecan easily be performed, and a period of time required for the robustspline filtration of measurement data can be shortened.

Further in this aspect of the present invention, in the measurement datainput step, the measurement data is preferably inputted with aprespecified space along the measurement route.

In this aspect of the present invention, not only the input data isinputted along a prespecified scanning direction, but also even whenmeasurement data obtained by scanning along a surface of a workpiece tobe measured is inputted, the measurement data is inputted with aprespecified space along the measurement route, a form changing point(such as a changing point from a straight portion to a circular portionor a boundary point of a step) can be grasped more accurately ascompared to a case where measurement data is inputted with aprespecified space in the x-axial direction. Namely, a mistake indetermination on a form of a workpiece to be measured can be prevented,which enables input of measurement data with higher reliability.

In this aspect of the present invention, the measurement data input steppreferably includes a step of deleting singular point data locallyisolated from the measurement data.

In this aspect of the present invention, for instance, locally andextremely isolated data (such as data obtained at a point and having anextremely different value from those obtained at adjoining points)included in the measurement data due to, for instance, generation ofstrong induction noise caused by a power machine in a plant as a noisesource can previously be deleted as clear singular point data, so thatthe reliability of the robust spline filtration is further improved.

The signal processing program in this aspect of the present inventionpreferably makes a computer execute the signal processing method.Further the recording medium according to the present inventionpreferably records therein the signal processing program in thecomputer-readable state. Further the signal processor according to thepresent invention preferably makes the computer execute the signalprocessing program.

With the configuration, by compiling a program so that each of the stepsdescribed above is executed by a computer incorporating a CPU (CentralProcessing Unit) or a memory (storage device), it is possible not onlyto adjust a weight and make determination concerning convergence of theweight, but also to easily change various parameters for deciding adifference according to a dimension or dimensions of measurement data.The program as described above may be installed in the computer bydirectly setting the recording medium with the program recorded therein,or a reader capable of reading out information from the recording mediumis externally connected to the computer so that the program can beinstalled in the computer with this reader. Further the program may befed to and installed in a computer via a communication line such as theInternet, LAN cable, telephone line, radio communication network or thelike.

In this aspect of the present invention, there is preferably provided asignal processing filter for executing signal processing by summingproducts for a prespecified distribution function in relation to digitalsignals at a specified dimension measured along a prespecifiedmeasurement route, and the distribution function is defined with asampling pitch along the prespecified route and a cut-off wavelengthalong the prespecified route, and further the signal processing filterpreferably has a filtration executing section for obtaining a filteroutput value for each component by multiplying components of the digitalsignal value with the distribution function.

With the configuration as described above, signal processing is executedfor each component of a digital signal value at a specified dimension,for instance, a two-dimensional or three-dimensional signal value. Inthis signal processing, the filtration executing section obtains afilter output value by multiplying an inputted digital signal by adistribution function and linearly summing the products. Thedistribution function is defined a sampling pitch and a cut-offwavelength each along a route for extracting a digital signal. Namely, asum of products of digital signals is computed through the distributionfunction by scanning along the route for extracting digital signals. Asa result, reliable signal processing can be executed along the route forextracting signals.

As described above, signal processing can be executed accurately notonly for one-dimensional data, but also for two- or three-dimensionaldigital data by executing signal processing for each component of adigital signal through the distribution function along the samplingroute. As a result, for instance, even when signal processing executedfor data concerning a form of a surface of a workpiece to be measured,form analysis can be made along the measurement route, so that changingpoints in the form can accurately be grasped, which enables furtherimprovement in precision of form analysis.

In this aspect, the signal processing filter preferably includes aweighting factor computing section for computing a weighting factor foreach of the digital signal values, and the filtration executing sectionpreferably executes an operation for summing products of thedistribution function with the weighting factor computed in theweighting factor computing section for each component of the digitalsignal.

With the configuration as described above, a weighting factor is set bythe weighting factor computing section for each digital signal. A weightfor each digital signal is adjusted according to this weighting factor,and further an operation for summing products by the distributionfunction is executed.

By adjusting a weighting factor for each of the digital signals, forinstance, influence by abnormal data can be excluded.

In this aspect of the present invention, the weighting factor computingsection preferably decide the weighting factor according to a differencebetween the digital signal value and the filter output value for thedigital signal value computed based on a sum of squares of eachcomponents of the digital signal value.

With the configuration as described above, the weighting factorcomputing section is required only to compute a digital signal value anda sum of squares of each component of the digital signal, so that thework load to the weighting factor computing section is reduced. Althoughit is possible to decide a weighting factor based on the shortestdistance between a digital signal and a curve obtained by filtration,but in that case the load for computing data at a higher dimensionsubstantially increases. With the present invention, however, a numberof computing steps is small, so that the output response speed cansubstantially be raised.

In this aspect, the weighting factor computing section re-computes theweighting factor using a filter output value from the filtrationexecuting section, and the signal processing filter has a convergencedetermination section for determining whether the weighting factor hasconverged or not based on a change rate of the weighting factor computedin the weighting factor computing section as well as of the weightingfactor computing in the previous step, and further the filtrationexecuting section outputs a filter output value based on the weightingfactor determined as converged by the convergence determination section.

With the configuration as described above, whether the signal processingis to be terminated or not is determined by the convergencedetermination section based on a change rate of a weighting factor. Whenit is determined that the weighting factor has converged, computing of aweighting factor is terminated, and a filter output is obtainedaccording to the converged weighting factor. As determination concerningconvergence is made on a change rate of a weighting factor, a result ofsignal processing with a converged weighting factor can be obtained forall digital signal values.

In this aspect of the present invention, signal processing can beexecuted to two- or three-dimensional digital signal values.

In this aspect, the distribution faction is preferably the Gaussiandistribution function.

With the configuration as described above, the Gaussian filter or therobust Gaussian filter can be constructed.

The distribution may be a box-type function constituting a movementaveraging filter.

In this aspect of the present invention, the signal processing filterincludes a weighting factor computing section for computing a weightingfactor for each of the digital signal values, and the weighting factorcomputing section preferably divides the digital signal to a pluralityof zones with a prespecified pitch along a measurement route, computesthe digital signal value and a median for a difference based on a sum ofsquares of components of each of the digital signal values, and decidesa weighting factor for each signal value through an equation fordeciding a weighting factor based on the median for this zone.

With the configuration as described above, as data in each zone isweighted according to the median data computed for the zone, a filteroutput well reflecting actual data for each zone can be obtained. Theequation for deciding a case based on a median computed for each zonemay be applied to all of signal values in the zone, or a portion ofsignal values in the zone. The zone pitch is, for instance,substantially equal to the cut-off wavelength.

This aspect of the present invention provides a signal processing methodfor executing signal processing by summing products for a prespecifieddistribution function for digital signal values at a specified dimensionobtained through measurement along a specified route, and thedistribution function is defined by sampling pitch along the specifiedroute and a cut-off wavelength along the specified route, and preferablyincludes a weighting factor computing step of computing a weightingfactor for each of the digital values, and a filtration executingsection for summing products of the weighting factor computed in theweighting factor computing step by the distribution function for eachcomponent.

With the configuration as described above, the advantages provided bythe invention described above can be achieved. Namely, by adjusting aweight for ach of the digital signal values, for instance, influence byabnormal data can be excluded.

In this aspect of the present invention, the weighting factor computingstep preferably decides the weighting factor based on a differencebetween the digital signal value and a sum of squares of each componentof the filter output value for the digital signal value.

With the configuration as described above, the advantages provided bythe invention described above can be achieved. It is required only tocompute a digital signal value and a sum of squares of each component ofa filter output value for the digital signal value, so that the workload for computing is reduced.

In this aspect of the present invention, the weighting factor computingstep preferably includes a convergence determination step in which theweighting factor is re-computed using a filter output value obtained inthe filtration executing section and determination concerningconvergence of the weighting factor is made based on a change rate ofthe weighting factor computed in the weighting factor computing step aswell as of the weighting factor computed in the previous step, and thefiltration executing step preferably outputs a filter output value usingthe weighting factor determined as converged in the convergencedetermination step.

With the configuration as described above, the advantages provided bythe invention described above can be achieved. Namely, as convergence ofa weighting factor is determined, it is not necessary to executeunnecessary repetitive computing, and further as convergence isdetermined based on a change rate of the weighting factor, so that anaccurately converged result can be obtained for all digital signalvalues.

The signal processing method in this aspect of the present inventionexecutes filtration for measurement data at a specified dimension, andincluding a measurement data input step of inputting the measurementdata along a measurement route; a selection step of selecting a weightedspline filter according to a type of the measurement data; aninitialization step of giving a weight for the measurement data in theform of a unit matrix to obtain an initial value of the spline filteroutput; an adjustment step of adjusting and deciding a weight for themeasurement data; a spline filter output computing step of obtaining aspline filter output by using the weight decided in the weightadjustment step; a convergence determination step of determining whetherthe weight has converged or not; and an output step of outputting aresult of signal processing based on the spline filter output, and thesignal processing method is characterized in that, when it is determinedin the convergence determination step that the weight has not converged,the weight adjustment step and spline filter output computing step arerepeated updating the weight to subject the measurement data to therobust spline filtration, and also in that the weight adjustment stepdivides the measurement data into a plurality of zones with aprespecified pitch along a route, computes a median for a differencebetween each measurement data and the spline curve computed by theequation for the weighted spline filtration, and decides a weight foreach data in the zone with the equation for deciding a weighting factorbased on the median for the zone.

With the present invention, an equation for weighted spline filtrationis selected, a spline curve reflecting spline filter outputs isrepeatedly computed by successively updating the weight based on theselected equation for spline filtration, so that the measurement datacan easily be subjected to the robust spline filtration providing aspline curve at the time of convergence of the weight as an filteroutput which is a result of signal processing. Because of the features,end-effects at a start point and at an end effect of the measurementdata can be prevented, and a form included in the measurement data canbe extracted without being affected by following capability for wavinesscomponents each having a long cycle nor by noise components included inthe measurement data. As a result, filtration well reflecting the actualform can be executed, and reliability of measurement data is furtherimproved.

A weight for data in each data is decided based on median data computedfor the data, so that a filter output well reflecting actual data foreach zone can be obtained. The equation for deciding a weight based on amedian for each zone may be applied to all signal values in the zone orto a portion of the signal values in the zone. The zone pitch is, forinstance, substantially equal to the cut-off wavelength.

The signal processing, signal processing method, signal processingprogram, recording medium, and measuring instrument according to thepresent invention are not limited to those described in the embodiments,reference technologies, and reference variants, and it is needless tosay that various modifications and changes are possible within the gistof the present invention.

For instance, the computing section 52 may be a computer including a CPU(Central Processing Unit) or a memory (storage device) for realizingeach of functions executed by the filtration executing section 57,weighting factor computing section 56, and convergence determiningsection 58. In this case, the signal processing program for making thecomputer execute the functions executed by the filtration executingsection 57, weighting factor computing section 56, and convergencedetermining section 58 may be installed via a communication network suchas the Internet or via a recording medium such as a CD-ROM or a memorycard. Further a reader for reading data stored in the recording mediummay be externally attached thereto. With the configuration as describedabove in which the computing section 52 is configured with a computerand signal processing is executed by software, some of parameters suchas a sampling pitch, a cut-off wavelength, and conditions forconvergence can easily be changed.

For instance, in the second reference technology, a weighting factor iscomputed by the Biweight method and the constant c is decided throughthe equation (9) according to the median data β, but the constant c maybe a prespecified fixed value.

Further the Beaton function may be employed in place of the adaptivetype Biweight method for computing a weighting factor.

The priority application Number JP2004-010921 upon which this patentapplication is based is hereby incorporated by reference.

1. A signal processor for filtering digital signal values in aprespecified dimension measured along a preset route, comprising: aweighting factor calculating section for calculating a weighting factorfor each of the digital signal values as well as for re-calculating andupdating the weighting factor; and a filtration executing section forexecuting filtration using the calculated weighting factor to obtain afilter output value for each digital signal value, wherein the weightingfactor calculating section comprises: a first zone generating sectionfor generating a plurality of first zones by dividing the digital signalvalues into a plurality of first zones along the preset route; and amedian calculating section for calculating a median for each of thefirst zones for a difference based on a sum of squares for respectivecomponents of the digital signal value and the filter output value forthe digital signal value, wherein the weighting factor calculatingsection calculates the weighting factor based on the median in each ofthe first zones for the digital signal values in the respective firstzones and updates the weighting factor.
 2. The signal processoraccording to claim 1, wherein the filtration executing section executesrobust spline filtration processing using a spline function.
 3. Thesignal processor according to claim 1, wherein the filtration executingsection executes robust Gaussian filtration processing using a Gaussianfunction.
 4. A signal processor for filtering digital signal values in aprespecified dimension measured along a preset route, comprising: aweighting factor calculating section for calculating a weighting factorfor each of the digital signal values as well as for recalculating andupdating the weighting factor; and a filtration executing section forexecuting filtration using the calculated weighting factor to obtain afilter output value for each digital signal value, wherein the weightingfactor calculating section comprises: a first zone generating sectionfor generating a plurality of first zones by dividing the digital signalvalue into a plurality of zones along the preset route; a second zonegenerating section for generating a plurality of second zonescorresponding to the first zones along the preset route of the digitalsignal value; and a median calculating section for calculating a medianfor a difference in accordance with a sum of squares for respectivecomponents of the digital signal value and the filter output value forthe digital signal value for the respective second zones, wherein theweighting factor calculating section calculates the weighting factorbased on the median for the second zones corresponding to the respectivefirst zones concerning the digital signal values in the respective firstzones and updates the weighting factor.
 5. The signal processoraccording to claim 4, wherein the second zone generating sectiongenerates the second zones each having a larger zone width as comparedto a width of the first zones.
 6. The signal processor according toclaim 5, wherein the first zone generating section generates the firstzones in which the digital signal values of adjacent first zones aresequentially connected to each other, and the second zone generatingsection generates the second zones including the digital signal valuesof the corresponding first zones and having portions in which adjacentsecond zones are overlapping with each other.
 7. The signal processoraccording to claim 4, wherein the filtration executing section executesrobust spline filtration processing using a spline function.
 8. Thesignal processor according to claim 4, wherein the filtration executingsection executes robust Gaussian filtration processing using a Gaussianfunction.
 9. A signal processing method for filtering digital signalvalues in a prespecified dimension measured along a preset route,comprising the steps of: scanning a surface of a workpiece along thepreset route; calculating a weighting factor for each of the digitalsignal values as well as for re-calculating and updating the weightingfactor; obtaining filter output values for the digital signal values byusing the calculated weighting factor; obtaining a spline curve from thefilter output values; and displaying the spline curve on a displaysection to determine the measurement of the surface of the workpiecealong the preset route, wherein calculating the weighting factorcomprises the steps of: generating a plurality of first zones bydividing the digital signal values into a plurality of first zones alongthe preset route; and calculating a median for a difference inaccordance with a sum of squares for respective components of thedigital signal value and the filter output value for the digital signalvalue, wherein the weighting factor calculation is based on the mediansfor the respective first zones concerning the digital signal values inthe respective first zones.
 10. The method according to claim 9, whereinthe method is implemented by a signal processing program that instructsa computer to execute the method steps for filtering digital signalvalues.
 11. The method according to claim 10, wherein the signalprocessing program is recorded on a recording medium.
 12. The signalprocessing method for filtering digital signal values in a prespecifieddimension measured along a preset route, comprising: scanning a surfaceof a workpiece along the preset route; calculating a weighting factorfor each of the digital signal values as well as for re-calculating andupdating the weighting factor; obtaining filter output values for thedigital signal values using the calculated weighting factor obtaining aspline curve from the filter output values; and displaying the splinecurve on a display section to determine the measurement of the surfaceof the workpiece along the preset route, wherein calculating theweighting factor comprises the steps of: generating a plurality of firstzones by dividing the digital signal values into a plurality of firstzones along the preset route; generating a plurality of second zonescorresponding to the first zones along the route of the digital signalvalue; and calculating a median for a difference in accordance with asum of squares for respective components of the digital signal value andthe filter output value for the digital signal value, wherein theweighting factor calculation for the digital signal values in each ofthe first zones is based on the median of the second zones correspondingto each of the first zones and updates the weighting factor.
 13. Themethod according to claim 12, wherein the method is implemented by aprogram that instructs a computer to execute the method steps forfiltering digital signal values.
 14. The method according to claim 13,wherein the program is recorded on a recording medium.
 15. A measuringinstrument comprising: a detector having a measuring section at an edgethereof for scanning a surface of a workpiece for measurement in one ofa contact state and a non-contact state; a moving unit for moving themeasuring section two-dimensionally or three-dimensionally; a positiondetecting section for outputting a measurement result obtained with themeasuring section as coordinate measurement data; a movement controlsection controlling movement of the measuring section, a signalprocessor for filtering digital signal values in a prespecifieddimension measured along a preset route, the processor comprising: aweighting factor calculating section for calculating a weighting factorfor each of the digital signal values as well as for re-calculating andupdating the weighting factor; and a filtration executing section forexecuting filtration using the calculated weighting factor to obtain afilter output value for each digital signal value, wherein the weightingfactor calculating section includes: a first zone generating section forgenerating a plurality of first zones by dividing the digital signalvalues into a plurality of first zones along the preset route; and amedian calculating section for calculating a median for each of thefirst zones for a difference based on a sum of squares for respectivecomponents of the digital signal value and the filter output value forthe digital signal value, the weighting factor calculating sectioncalculating the weighting factor based on the median in each of thefirst zones for the digital signal values in the respective first zonesand updates the weighting factor; and an output section for outputting aresult that was filtered through the signal processor.
 16. A measuringinstrument comprising: a detector having a measuring section at an edgethereof for scanning a surface of a workpiece for measurement in one ofa contact state and a non-contact state; a moving unit for moving themeasuring section two-dimensionally or three-dimensionally; a positiondetecting section for outputting a measurement result obtained with themeasuring section as coordinate measurement data; a movement controlsection controlling movement of the measuring section, a signalprocessor for filtering digital signal values in a prespecifieddimension measured along a preset route, the processor comprising: aweighting factor calculating section for calculating a weighting factorfor each of the digital signal values as well as for recalculating andupdating the weighting factor; and a filtration executing section forexecuting filtration using the calculated weighting factor to obtain afilter output value for each digital signal value, wherein the weightingfactor calculating section includes: a first zone generating section forgenerating a plurality of first zones by dividing the digital signalvalue into a plurality of first zones along the preset route; a secondzone generating section for generating a plurality of second zonescorresponding to the first zones along the preset route of the digitalsignal value; and a median calculating section for calculating a medianfor a difference in accordance with a sum of squares for respectivecomponents of the digital signal value and the filter output value forthe digital signal value for the respective second zones, the weightingfactor calculating section calculating the weighting factor based on themedian for the second zones corresponding to the respective first zonesconcerning the digital signal values in the respective first zones andupdates the weighting factor; and an output section for outputting aresult that was filtered through the signal processor.